An application of the continuous Steiner symmetrization to Blaschke-Santaló diagrams

In this paper we consider the so-called procedure of {it Continuous Steiner Symmetrization}, introduced by Brock in~cite{bro95,bro00}. It transforms every open set $OmegacompR^d$ into the ball keeping the volume fixed and letting the first eigenvalue and the torsional rigidity respectively decrease and increase. While this does not provide, in general, a $gamma$-continuous map $tmapstoOmega_t$, it can be slightly modified so to obtain the $gamma$-continuity for a $gamma$-dense class of domains $Omega$, namely, the class of polyhedral sets in $R^d$. This allows to obtain a sharp characterization of the Blaschke-Santal'o diagram of torsion and eigenvalue.